Math 8302
Spring 2020
Location and time: Vincent Hall 1, MWF
1:25pm-2:15pm
Text: Foundations of Differentiable Manifolds and Lie Groups
by Frank Warner.
Additional text: Differential Forms in Algebraic Topology
by R. Bott and L. Tu.
Lecturer: Tian-Jun Li, Vincent
Hall 260,
(612) 625-2036
Email: lixxx248@umn.edu
URL:
http://www.math.umn.edu/~lixxx248
Office hours: 12:15-1:15, MF.
Course Content
We will cover Chapters 1, 2, 4 and selected topics in Chapters 3, 5 and 6.
Homework
There will be 7 or 8 homework problem sets.
Homework 1, due Feb 7 (Friday).
1-4. Exercises in Chapter 1: 1.1, 1.2, 1.5, 1.24 (a, b, d)
5. Show that R^k and R^l are not diffeomorphic if k and l are different.
6. Show that the tangent bundle of S^1 is diffeomorphic to S^1 x R.
7. Show that there is no immersion from S^1 x S^1 to R^2.
8. Construct an immersion from R^2 to R^2 which is not a diffeomorphism onto its image.
9-10. Exercises in Chapter 1: 1.9, 1.10.
Solutions to Hw1
Homework 2, due Feb 21 (Friday).
1-6. Exercises in Chapter 1: 1.12, 1.13, 1.15, 1.17, 1.18, 1.24(c)
7. Show that the set of 2 by 2 matrices with rank 1 is a submanifold of R^4 with codimension 1.
8. Exercise 1.21. Moreover, determine whether the skew line is an embedding.
9. In the setting of Theorem 1.38, show that the tangent space of P at m is mapped by d i to the kernel of d\psi
at m.
Solutions to Hw2
Homework 3, due March 18 (Wednesday).
1-6. Exercises in Chapter 2: 2.3, 2.4, 2.11, 2.16
More to come
Tests.
There are two in class tests and a take home final.
Test 1 on Friday, 2/28. It covers Chapter 1.
Test 2 on Monday, 4/3. It covers Chapter 2.
Grading
Each test accounts for %15 and the take home final accounts for %20, and homework accounts for %50.
Test 1 on 2/28 (Friday) and test 2 on 4/3 (Friday).